How to cite from a paper using a different citation style?

Echoing the other good comments and answer, I'd encourage you to think in terms of "forthrightness", rather than compliance to semi-random formal criteria. Think of the function of your citations, and thinking of the idealized version of scholarship and such. The goal is not conformity to style guides, but useful contributions to human knowledge, blah-blah-blah, ;) But, yes, that should be the way to think about such questions.

To my taste, citations in-line should be sufficient so that the reader does not have to "flip to the end" to see what "[46]" is. The space spent on fuller references is a good investment!!!

That is, yet again, if one thinks that the function of one's writing is to help the reader, all in-line references would be optimally helpful, as opposed to "proper-and-possibly-unhelpful".

Why not rephrase it, and quote each source properly? Like this:

McKee (2011), Busby (2008) and in Barker, Grout, & Crowe (2011) wrote that “Malnutrition has often been referred to as 'the skeleton in the hospital closet', as it is often overlooked, under-diagnosed, and untreated". To this, in Barker, Grout & Crowe added that "Despite this, the negative consequences of malnutrition have been widely reported...".

In a way, you are quoting McKee at al., so it is only proper to cite them properly, and thus have them in your references.